Problem 15
Question
Find the \(x\) - and \(y\) -intercepts of the rational function. $$r(x)=\frac{x^{2}-9}{x^{2}}$$
Step-by-Step Solution
Verified Answer
x-intercepts: (3, 0) and (-3, 0); No y-intercept.
1Step 1: Identify the Rational Function
The given rational function is \( r(x) = \frac{x^2 - 9}{x^2} \). Our task is to find the \(x\)- and \(y\)-intercepts.
2Step 2: Find the x-intercepts
To find the \(x\)-intercepts, set the numerator of the rational function equal to zero and solve for \(x\). Thus, set \(x^2 - 9 = 0\). This results in \(x^2 = 9\), which gives \(x = \pm 3\). Therefore, the \(x\)-intercepts are at \((3, 0)\) and \((-3, 0)\).
3Step 3: Find the y-intercept
To find the \(y\)-intercept, evaluate the function at \(x = 0\). Substitute \(x = 0\) into \( r(x) = \frac{x^2 - 9}{x^2} \), yielding \( r(0) = \frac{0 - 9}{0} \). This is undefined, which means there is no \(y\)-intercept.
Key Concepts
Understanding x-intercepts in Rational FunctionsGrasping the Concept of y-interceptsIdentifying Undefined Values in Rational Functions
Understanding x-intercepts in Rational Functions
When we talk about the *x*-intercepts of a rational function, we're essentially identifying where the graph of the function crosses the x-axis. At these points, the value of the function, or the output, is zero. Therefore, to find the *x*-intercepts, you set the numerator of the rational function equal to zero since a fraction is zero only when its numerator is zero.
For example, in the function \(r(x) = \frac{x^2 - 9}{x^2}\), you identify the *x*-intercepts by solving \(x^2 - 9 = 0\). This step leads to \(x^2 = 9\), giving solutions \(x = 3\) and \(x = -3\). Thus, the *x*-intercepts are at \((3, 0)\) and \((-3, 0)\).
This tells us the rational function crosses the x-axis at these points, expressing where the function yields a zero value for a non-zero denominator.
For example, in the function \(r(x) = \frac{x^2 - 9}{x^2}\), you identify the *x*-intercepts by solving \(x^2 - 9 = 0\). This step leads to \(x^2 = 9\), giving solutions \(x = 3\) and \(x = -3\). Thus, the *x*-intercepts are at \((3, 0)\) and \((-3, 0)\).
This tells us the rational function crosses the x-axis at these points, expressing where the function yields a zero value for a non-zero denominator.
Grasping the Concept of y-intercepts
The *y*-intercept of a rational function is where the graph intersects the y-axis. At this point, the *x*-coordinate is zero, and so to find the *y*-intercept, you substitute \(x = 0\) into the function.
However, sometimes, as in our example \(r(x) = \frac{x^2 - 9}{x^2}\), calculating \(r(0)\) leads to an undefined expression because dividing by zero is mathematically undefined. Specifically, \(r(0) = \frac{0 - 9}{0}\) demonstrates a division by zero situation, indicating that there's no *y*-intercept.
When this happens, it usually means the graph does not cross the y-axis, which is important for understanding the behavior and limits of the function at that point. This scenario should remind us of the critical role denominators play in defining a rational function's behavior.
However, sometimes, as in our example \(r(x) = \frac{x^2 - 9}{x^2}\), calculating \(r(0)\) leads to an undefined expression because dividing by zero is mathematically undefined. Specifically, \(r(0) = \frac{0 - 9}{0}\) demonstrates a division by zero situation, indicating that there's no *y*-intercept.
When this happens, it usually means the graph does not cross the y-axis, which is important for understanding the behavior and limits of the function at that point. This scenario should remind us of the critical role denominators play in defining a rational function's behavior.
Identifying Undefined Values in Rational Functions
Undefined values in rational functions occur when the denominator equals zero as division by zero is undefined in mathematics. This results in points where the function 'breaks', often represented as vertical asymptotes or gaps in the graph.
In the given function \(r(x) = \frac{x^2 - 9}{x^2}\), the denominator \(x^2\) becomes zero when \(x = 0\). Thus, \(x=0\) is an undefined value.
To fully grasp what this means for the function, consider the implications:
In the given function \(r(x) = \frac{x^2 - 9}{x^2}\), the denominator \(x^2\) becomes zero when \(x = 0\). Thus, \(x=0\) is an undefined value.
To fully grasp what this means for the function, consider the implications:
- This function is undefined at \(x = 0\), so there's a discontinuity or a break in the graph along the x-axis.
- Such undefined points must be carefully identified as they impact the function's domain and its graphical representation.
Other exercises in this chapter
Problem 14
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